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DOE OSTI2026

Nonlinear optimal recovery in Hilbert spaces

Here, this paper investigates solution strategies for nonlinear problems in Hilbert spaces, such as nonlinear partial differential equations (PDEs) in Sobolev spaces, when only finite measurements are available. We formulate this as a nonlinear optimal recovery problem, establishing its well-posedness and proving its convergence to the true solution as the number of measurements increases. However, the resulting formulation might not have a finite-dimensional solution in general. We thus present a sufficient condition for the finite dimensionality of the solution, applicable to problems with well-defined point evaluation measurements. To address the broader setting, we introduce a relaxed nonlinear optimal recovery and provide a detailed convergence analysis. An illustrative example is given to demonstrate that our formulations and theoretical findings offer a comprehensive framework for solving nonlinear problems in infinite-dimensional spaces with limited data.

convergence↗

DOE OSTI2022

Automatic Differentiation of C++ Codes on Emerging Manycore Architectures with Sacado

Automatic differentiation (AD) is a well-known technique for evaluating analytic derivatives of calculations implemented on a computer, with numerous software tools available for incorporating AD technology into complex applications. However, a growing challenge for AD is the efficient differentiation of parallel computations implemented on emerging manycore computing architectures such as multicore CPUs, GPUs, and accelerators as these devices become more pervasive. In this work, we explore forward mode, operator overloading-based differentiation of C++ codes on these architectures using the widely available Sacado AD software package. In particular, we leverage Kokkos, a C++ tool providing APIs for implementing parallel computations that is portable to a wide variety of emerging architectures. Here we describe the challenges that arise when differentiating code for these architectures using Kokkos, and two approaches for overcoming them that ensure optimal memory access patterns as well as expose additional dimensions of fine-grained parallelism in the derivative calculation. We describe the results of several computational experiments that demonstrate the performance of the approach on a few contemporary CPU and GPU architectures. We then conclude with applications of these techniques to the simulation of discretized systems of partial differential equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

hIPPYlib-MUQ: A Bayesian Inference Software Framework for Integration of Data with Complex Predictive Models under Uncertainty

Bayesian inference provides a systematic framework for integration of data with mathematical models to quantify the uncertainty in the solution of the inverse problem. However, the solution of Bayesian inverse problems governed by complex forward models described by partial differential equations (PDEs) remains prohibitive with black-box Markov chain Monte Carlo (MCMC) methods. We present hIPPYlib-MUQ, an extensible and scalable software framework that contains implementations of state-of-the art algorithms aimed to overcome the challenges of high-dimensional, PDE-constrained Bayesian inverse problems. These algorithms accelerate MCMC sampling by exploiting the geometry and intrinsic low-dimensionality of parameter space via derivative information and low rank approximation. The software integrates two complementary open-source software packages, hIPPYlib and MUQ. hIPPYlib solves PDE-constrained inverse problems using automatically-generated adjoint-based derivatives, but it lacks full Bayesian capabilities. MUQ provides a spectrum of powerful Bayesian inversion models and algorithms, but expects forward models to come equipped with gradients and Hessians to permit large-scale solution. By combining these two complementary libraries, we created a robust, scalable, and efficient software framework that realizes the benefits of each and allows us to tackle complex large-scale Bayesian inverse problems across a broad spectrum of scientific and engineering disciplines. To illustrate the capabilities of hIPPYlib-MUQ, we present a comparison of a number of MCMC methods available in the integrated software on several high-dimensional Bayesian inverse problems. These include problems characterized by both linear and nonlinear PDEs, various noise models, and different parameter dimensions. The results demonstrate that large (~ 50×) speedups over conventional black box and gradient-based MCMC algorithms can be obtained by exploiting Hessian information (from the log-posterior), underscoring the power of the integrated hIPPYlib-MUQ framework.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

A Spatiotemporal Prediction Model for Regional Scheduling of Shared Bicycles Based on the INLA Method

Dock-less bicycle-sharing programs have been widely accepted as an efficient mode to benefit health and reduce congestions. And modeling and prediction has always been a core proposition in the field of transportation. Most of the existing demand prediction models for shared bikes take regions as research objects; therefore, a POI-based method can be a beneficial complement to existing research, including zone-level, OD-level, and station-level techniques. Point of interest (POI) is the location description of spatial entities, which can reflect the cycling route characteristics for both commuting and noncommuting trips to a certain extent, and is also the main generating point and attraction point of shared-bike travel flow. In this study, we make an effort to model a POI-level cycling demand with a Bayesian hierarchical method. The proposed model combines the integrated nested Laplace approximation (INLA) and random partial differential equation (SPDE) to cope with the huge computation in the modeling process. In particular, we have adopted the dock-less bicycle-sharing rental records of Mobike as a case study to validate our method; the study area was one of the fastest growing urban districts in Shanghai in August 2016. The operation results show that the method can help better understand, measure, and characterize spatiotemporal patterns of bike-share ridership at the POI level and quantify the impact of the spatiotemporal effect on bicycle-sharing use.

Yu, Zhuoran↗

DOE OSTI2020

MEUMAPPS (Microstructure Evolution Using Massively Parallel Phase-field Simulations)

The software “MEUMAPPS” is a h high-performance computing code used to simulate microstructure evolution associated with diffusional solid-state transformations in structural alloys. Understanding microstructure evolution during thermo-mechanical processing of structural alloys is the first step towards designing and processing alloys for specific applications by meeting property requirements demanded by the application. In this instance, the code is an important component in predicting processing-structure linkages during additive manufacturing of structural alloys as a function of processing parameters and alloy composition used in an additive manufacturing process. The code provides a detailed three-dimensional distribution of different types of phases / constituents and their morphologies, as well as the compositions of these phases that make up the microstructure. The microstructure is simulated by solving the governing partial differential equations using a Fourier Spectral Method.

Radhakrishnan, Balasubram↗

DOE OSTI2022

PsDNS

PsDNS is a Python package which makes it easy to solve partial differential equation using a pseudo-spectral method. The code uses Python classes and Numpy-like arrays to provide a simple interface with which users can construct equations, integrators, and diagnostics. It also includes implementation for common problems, in particular, the incompressible Navier-Stokes equation in a periodic domain, which is commonly used as a benchmark problem for turbulence research. PsDNS uses MPI to allow massively parallel computation for large problems.

Israel, Daniel↗

DOE OSTI2022

DD-AADL

Data Driven Anderson Acceleration for Physics Informed Neural Networks to solve high dimensional partial differential equations

Lupo Pasini, Massimiliano [Oak Ridge National Lab.↗

DOE OSTI2022

Neural Networks for Supercapacitor Modeling

This software implements a one-dimensional model of a supercapacitor and calibrates it to experimental data. A system of coupled partial differential equation models the salt concentration and the ionic and electronic potentials. The term in the equation modeling the capacitance contains a neural network, whose parameters are optimized to fit the provided experimental data. The software is written in python using the open-source library JAX to calculate the gradients needed for the optimization.

Salazar de Troya, Miquel↗

DOE OSTI2021

CARDINAL

Cardinal is a code that automates the workflow for iteratively solving neutral particle transport via the OpenMC code and computational fluid dynamics via the NekRS code within the Multiphysics Object Oriented Simulation Environment (MOOSE), a finite element framework for solving general partial differential equations. Cardinal establishes a mapping between the Monte Carlo geometry representation used in OpenMC and the spectral element mesh used in NekRS to other physics applications built upon the MOOSE framework, allowing high-resolution particle transport and fluid dynamics physics feedback to the MOOSE application ecosystem. Heat generation rates as solved by OpenMC are used as heat sources in MOOSE, while the temperature and density fields from NekRS are used as conjugate heat transfer boundary conditions and/or source fields in MOOSE. OpenMC, Nek5000, and MOOSE are open source, community-developed simulation codes.

NOVAK, APRILJEAN RODGERS↗

DOE OSTI2023

PINNSTRIPES (Physics-Informed Neural Network SurrogaTe for Rapidly Identifying Parameters in Energy Systems) [SWR-22-12]

Energy systems models typically take the form of complex partial differential equations which make multiple forward calculations prohibitively expensive. Fast and data-efficient construction of surrogate models is of utmost importance for applications that require parameter exploration such as design optimization and Bayesian calibration. In presence of a large number of parameters, surrogate models that capture correct dependencies may be difficult to construct with traditional techniques. The issue is addressed here with the formulation of the surrogate model constructed via Physics-Informed Neural Networks (PINN) which capture the dependence with respect to the parameters to estimate, while using a limited amount of data. Since forward evaluations of the surrogate model are cheap, parameter exploration is made inexpensive, even when considering a large number of parameters.

Hassanaly, Malik↗

DOE OSTI2023

LANL contribution to ryujin, an open source finite element solver

Ryujin (https://github.com/conservation-laws/ryujin) is a high-performance finite-element software for solving mathematical partial differential equations (PDEs) with dominant hyperbolic structures. The author of this request, Eric Tovar, is using Ryujin as a high-performance tool for his Mark Kac postdoctoral fellowship research at LANL. Eric would like to contribute openly to the ryujin software without changing its core functionality. This includes: (i) bug fixes; (ii) re-organization of code for performance and syntactic updates including documentation; (iii) implementation of new PDE numerical methods that align with the core solver; (iv) implementation of new initial state configurations for target applications.

Tovar, Eric↗

DOE OSTI2022

Marmot V2

MARMOT is a robust numerical tool for mesoscale modeling of fuel performance developed under the NEAMS Fuels technical area to predict the coevolution of microstructure and properties in fuel and cladding materials. MARMOT accomplishes this using the phase field method coupled with finite strain mechanics and heat conduction. MARMOT is based on the open source Multiphysics Object-Oriented Simulation Environment (MOOSE) and solves the coupled partial differential equations defining the physics using the finite element method. MARMOT is being developed in order to facilitate the development of improved materials models for fuel performance, but it is also being developed as a powerful tool in and of itself for the simulation of mesoscale fuel performance.

Aagesen, LarryK.↗

DOE OSTI2024

AMReX v2024

The software framework, AMReX, supports the development of block-structured adaptive mesh refinement (AMR) algorithms for solving systems of partial differential equations. AMR reduces the computational cost and memory footprint compared to a uniform mesh while preserving the essential local descriptions of different physical processes in complex multiphysics algorithms. AMR uses a hierarchical representation of the solution at multiple levels of resolution where the solution on each level is defined on the union of data containers at that resolution. These data containers, which represent the solution over a logically rectangular subregion of the domain, can contain field data defined on a mesh, Lagrangian particles or combinations of both. In addition to these basic data types, AMReX supports a multilevel embedded boundary representation of complex geometry; linear solvers for cell-centered and nodal data; asynchronous I/O in a native format readable by ParaView, VisIt and yt; and interfaces to hypre and PETSc solvers. AMReX enables applications to run on distributed memory architectures with multicore CPUs and with GPU accelerators. AMReX uses a lightweight abstraction layer that effectively hides the details of the architecture from the application. The framework currently supports CUDA, HIP and SYCL for GPU acceleration and OpenMP for multi-core CPU architectures.

Almgren, Ann↗

DOE OSTI2023

Canonical and Noncanonical Hamiltonian Operator Inference

SAND2024-08577O Canonical and Noncanonical Hamiltonian Operator Inference is research code that applies old and new model reduction algorithms to well-known and publicly available "toy" problems. It applies operator inference (OpInf) methods for model reduction to several benchmark hyperbolic partial differential equations: linear wave, Korteweg-de Vries, Benjamin-Bona-Mahoney, and linear elasticity. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

SciDAC↗

DOE OSTI2024

Swift

Swift is a fast Fourier transform based spectral solver based on the MOOSE framework. It supports GPU accelerated semi-implicit solves of partial differential equations, such as those used for phase field mesoscale microstructure evolution.

Schwen, Daniel [Idaho National Laboratory (INL), I↗

DOE OSTI2025

Generic Discretization Library

The GenDiL library is a collection of C++ software abstractions designed to discretize and solve partial differential equations (PDEs) for high-performance computing (HPC) applications. Its primary focus is on modern C++ generic programming, which helps ensure portability across various hardware architectures. The central idea behind the library is to provide building blocks for numerical algorithms-such as discretization methods and iteration patterns-so that domain experts can focus on the math, rather than the low-level details of hardware or implementation. By defining abstractions for data types, iteration over computational grids, and scheduling of operations, the library isolates the high-level PDE algorithms from the platform-specific optimizations needed to achieve efficient performance.

Dudouit, Yohann [Lawrence Livermore National Labor↗

DOE OSTI2025

The Method of Finite Averages

The Method of Finite Averages (MoFA) is a rigorous multiscale modeling methodology for efficiently modeling multi-physical phenomena in heterogeneous porous media. The code developed in this project aims to perform the numerical calculations required to formulate, implement, and verify MoFA models for Earth and Energy systems (i.e., model verification refers to performing fully-resolved simulations of the systems and comparing their results to those of the models). In general, MoFA transforms partial differential equations (PDEs) describing the fine-scale physics of a system into coupled ordinary differential equations (ODEs)---in time---that describe the coarse-scale---or "average"---physical behaviors of the system. This transformation significantly expedites system simulation, as the coarse-scale ODEs involve vastly fewer degrees of freedom than the fine-scale PDEs. The code developed under this project will allow users to 1.) generate system geometries and numerical meshes, 2.) solve the PDE and ODE systems required for MoFA model formulation and implementation, 3.) solve the PDE systems required to obtain fully-resolved simulation results for model verification, and 4.) compare and plot results (e.g., the model and fully-resolved simulation solutions, the error between the solutions, etc.).

Pietrzyk, KyleM [Lawrence Livermore National Labor↗

DOE OSTI2026

Fortran mimetic abstraction language (Formal) v0.1.

The Fortran mimetic abstraction language ("Formal") is a domain-specific language (DSL) embedded in Fortran 202Y [1]. Formal provides novel software abstractions for simulating phenomena governed by the partial differential equations (PDEs) of vector and tensor calculus. Such equations model an extremely broad set of physical phenomena, ranging from atmospheric winds to light propagation. Formal's data structures and algorithms mimic in form and behavior continuous functions and operators. Formal supports these mathematical constructs using mimetic discretizations that define a discrete calculus satisfying various tensor calculus theorems, thereby ensuring high-fidelity representations of the physics being modeled. [2] Formal 0.1.0 also lays a foundation for the future use of Fortran 202Y type-safe templates to facilitate the formal verification of tensor contractions in computational physics and artificial intelligence [3]. [1] "Fortran 202Y" is Fortran standard committee's informal designation for the next Fortran revision, which will likely be "Fortran 2028". [2] Corbino, J. and Castillo, J. (2020) Journal of Computational and Applied Mathematics, https://doi.org/10.1016/j.cam.2019.06.042. [3] Haveraaen, M., Järvi, J., & Rouson, D. (2019). Reflecting on Generics for Fortran. https://j3-fortran.org/doc/year/19/19-188.pdf.

Rouson, Damian [Lawrence Berkeley National Laborat↗